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January  2018, 14(1): 81-103. doi: 10.3934/jimo.2017038

## Modeling and computation of water management by real options

 1 Coordinated Innovation Center for Computable Modeling in Management Science, Tianjin University of Finance and Economics, Tianjin, 300222, China 2 School of Mathematics and Statistics, Zhengzhou University, Zhengzhou 450001, China

* Corresponding author: Shuhua Zhang

Received  April 2016 Revised  February 2017 Published  January 2018 Early access  April 2017

Fund Project: This project was supported in part by the National Basic Research Program (2012CB955804), the Major Research Plan of the National Natural Science Foundation of China (91430108), the National Natural Science Foundation of China (11171251), and the Major Program of Tianjin University of Finance and Economics (ZD1302).

It becomes increasingly important to manage water and improve the efficiency of irrigation under higher temperatures and irregular precipitation patterns. The choice of investment in water saving technologies and its timing play key roles in improving efficiency of water use. In this paper, we use a real option approach to establish a model to handle future uncertainties about the water price. In addition, to match the practical situation, the expiration of the real option is considered to be finite in our model, such that it is difficult to solve the model. Therefore, we reformulate the problem into a linear parabolic variational inequality (Ⅵ) and develop a power penalty method to solve it numerically. Thus, a nonlinear partial differential equation (PDE) is obtained, which is shown to be uniquely solvable and the solution of the nonlinear PDE converges to that of the Ⅵ at the rate of $O(λ^{-\frac{k}{2}})$ with $λ$ being the penalty number. Furthermore, a so-called fitted finite volume method is proposed to solve the nonlinear PDE. Finally, several numerical experiments are performed. It is shown that the subjective discount rate will affect the investment boundary mostly, and the flexibility to suspend operation will enlarge the investment region.

Citation: Shuhua Zhang, Xinyu Wang, Hua Li. Modeling and computation of water management by real options. Journal of Industrial and Management Optimization, 2018, 14 (1) : 81-103. doi: 10.3934/jimo.2017038
##### References:
 [1] M. Akinlar, Application of a finite element method for variational inequalities, Journal of Inequalities and Applications, 2013 (2013), 6pp.  doi: 10.1186/1029-242X-2013-45. [2] R. Bagatin, J. Klemes, A. Reverberi and D. Huisingh, Conservation and improvements in water resource management: A global challenge, Journal of Cleaner Production, 77 (2014), 1-9.  doi: 10.1016/j.jclepro.2014.04.027. [3] J. Bosch, M. Stoll and P. Benner, Fast solution of Cahn-Hilliard variational inequalities using implicit time discretization and finite elements, Journal of Computational Physics, 262 (2014), 38-57.  doi: 10.1016/j.jcp.2013.12.053. [4] C. Boehm and M. Ulbrich, A semi-smooth Newton-CG method for constrained parameter identification in seismic tomography, SIAM Journal on Scientific Computing, 37 (2015), 334-364.  doi: 10.1137/140968331. [5] N. Buong and N. Anh, An implicit iteration method for variational inequalities over the set of common fixed points for a finite family of nonexpansive mappings in Hilbert spaces, Fixed Point Theory and Applications, 1 (2011), Art. ID 276859, 10 pp. [6] J. Carey and D. Zilberman, A model of investment under uncertainty: Modern irrigation technology and emerging markets in water, American Journal of Agricultural Economics, 84 (2002), 171-183.  doi: 10.1111/1467-8276.00251. [7] S. Chang, J. Wang and X. Wang, A fitted finite volume method for real option valuation of risks in climate change, Computers and Mathematics with Applications, 70 (2015), 1198-1219.  doi: 10.1016/j.camwa.2015.07.003. [8] S. Chang and X. Wang, Modelling and computation in the valuation of carbon derivatives with stochastic convenience yields, Plos One, 10 (2015), e0125679.  doi: 10.1371/journal.pone.0125679. [9] S. Chang, X. Wang and Z. Wang, Modeling and computation of transboundary industrial pollution with emissions permits trading by stochastic differential game, PLoS ONE, 10 (2015), e0138641.  doi: 10.1371/journal.pone.0138641. [10] L. Chorn and S. Shokhor, Real options for risk management in petrolem development investments, Energy Economics, 28 (2006), 489-505. [11] B. Diomande and A. Zalinescu, Maximum principle for an optimal control problem associated to a stochastic variational inequality with delay, Electronic Journal of Probability, 20 (2014), 1-35.  doi: 10.1214/EJP.v20-2741. [12] A. Dixit and R. Pindyck, Investment under Uncertainty, Princeton University Press, Princeton, 1994. [13] R. France, Exploring the bonds and boundaries of water management in a global context, Journal of Cleaner Production, 60 (2013), 1-3.  doi: 10.1016/j.jclepro.2013.07.004. [14] W. Han and B. Reddy, On the finite element method for mixed variational inequalities arising in elastoplasticity, SIAM Journal on Numerical Analysis, 32 (1995), 1778-1807.  doi: 10.1137/0732081. [15] Y. He, Real Options in the Energy Markets, Ph. D Thesis, University of Twente, 2007. [16] C. Huang, C. Hung and S. Wang, A fitted finite volume method for the valuation of options on assets with stochastic volatilities, Computing, 77 (2006), 297-320.  doi: 10.1007/s00607-006-0164-4. [17] C. Huang and S. Wang, A power penalty approach to a nonlinear complementarity problem, Operations Research Letters, 38 (2010), 72-76.  doi: 10.1016/j.orl.2009.09.009. [18] K. Ito and K. Kunisch, Parabolic variational inequalities: The Lagrange multiplier approach, J. Math. Pures Appl., 85 (2006), 415-449.  doi: 10.1016/j.matpur.2005.08.005. [19] L. Kobari, S. Jaimungal and Y. Lawryshyn, A real options model to evaluate the effect of environmental policies on the oil sands rate of expansion, Energy Economics, 45 (2014), 155-165.  doi: 10.1016/j.eneco.2014.06.010. [20] R. Leveque, Finite Volume Methods for Hyperbolic Problems, Cambridge University Press, Cambridge, 2002.  doi: 10.1017/CBO9780511791253. [21] J. Liu, L. Mu and X. Ye, An adaptive discontinuous finite volume method for elliptic problems, Journal of Computational and Applied Mathematics, 235 (2011), 5422-5431.  doi: 10.1016/j.cam.2011.05.051. [22] A. McClintock, Investment in Irrigation Technology: Water Use Change, Public Policy and Uncertainty, Cooperative Research Centre for Irrigation Futures, Technical Report, 2014. [23] D. Pimentel, Water resources: Agriculture, the environment, and society, BioScience, 47 (1997), 97-106.  doi: 10.2307/1313020. [24] J. Reyes and K. Kunisch, A semi-smooth Newton method for regularized state-constrained optimal control of the Navier-Stokes Equations, Computing, 78 (2006), 287-309.  doi: 10.1007/s00607-006-0183-1. [25] J. Reyes and M. Hintermuller, A duality based semismooth Newton framework for solving variational inequalities of the second kind, Interfaces and Free Boundaries, 13 (2011), 437-462.  doi: 10.4171/IFB/267. [26] P. Samuelson, Proof that properly anticipated prices fluctuate randomly, The World Scientific Handbook of Futures Markets, 6 (2015), 25-38.  doi: 10.1142/9789814566926_0002. [27] S. Wang, A novel fitted finite volume method for the Black-Scholes equation governing option pricing, IMA Journal of Numerical Analysis, 24 (2004), 699-720.  doi: 10.1093/imanum/24.4.699. [28] Y. Wang, X. Chang, Z. Chen, Y. Zhong and T. Fan, Impact of subsidy policies on recycling and remanufacturing using system dynamics methodology: a case of auto parts in China, Journal of Cleaner Production, 74 (2014), 161-171.  doi: 10.1016/j.jclepro.2014.03.023. [29] T. Wang and R. Neufville, Building real options into physical systems with stochastic mixed-integer programming, In 8th Annual Real Options International Conference, (2004), 23-32. [30] G. Wang and X. Yang, The regularization method for a degenerate parabolic variational inequality arising from American option valuation, International Journal of Numerical Analysis and Modeling, 5 (2008), 222-238. [31] S. Wang and X. Yang, A power penalty method for linear complementarity problems, Operations Research Letters, 36 (2008), 211-214.  doi: 10.1016/j.orl.2007.06.006. [32] S. Wang, X. Yang and K. Teo, Power penalty method for a linear complementarity problem arising from American option valuation, Journal of Optimization Theory and Applications, 129 (2006), 227-254.  doi: 10.1007/s10957-006-9062-3. [33] S. Wang, S. Zhang and Z. Fang, A superconvergent fitted finite volume method for BlackScholes equations governing European and American option valuation, Numerical Methods for Partial Differential Equations, 31 (2015), 1190-1208.  doi: 10.1002/num.21941. [34] A. Wasylewicz, Analysis of the power penalty method for American options using viscosity solutions, Thesis, University of Oslo, 2008. [35] S. Xie, H. Xu and H. Huang, Some iterative numerical methods for a kind of system of mixed nonlinear variational inequalities, Journal of Mathematics Research, 6 (2014), 65-69.  doi: 10.5539/jmr.v6n1p65. [36] A. Zalinescu, Stochastic variational inequalities with jumps, Stochastic Processes and their Applications, 124 (2014), 785-811.  doi: 10.1016/j.spa.2013.09.005. [37] S. Zhang, X. Wang and A. Shananin, Modeling and computation of mean field equilibria in producers' game with emission permits trading, Communications in Nonlinear Science and Numerical Simulation, 37 (2016), 238-248.  doi: 10.1016/j.cnsns.2016.01.020. [38] K. Zhang, S. Wang, X. Yang and K. Teo, A power penalty approach to numerical solutions of two-asset American options, Numerical Mathematics: Theory, Method and Applications, 2 (2009), 202-233. [39] [40] [41]

show all references

##### References:
 [1] M. Akinlar, Application of a finite element method for variational inequalities, Journal of Inequalities and Applications, 2013 (2013), 6pp.  doi: 10.1186/1029-242X-2013-45. [2] R. Bagatin, J. Klemes, A. Reverberi and D. Huisingh, Conservation and improvements in water resource management: A global challenge, Journal of Cleaner Production, 77 (2014), 1-9.  doi: 10.1016/j.jclepro.2014.04.027. [3] J. Bosch, M. Stoll and P. Benner, Fast solution of Cahn-Hilliard variational inequalities using implicit time discretization and finite elements, Journal of Computational Physics, 262 (2014), 38-57.  doi: 10.1016/j.jcp.2013.12.053. [4] C. Boehm and M. Ulbrich, A semi-smooth Newton-CG method for constrained parameter identification in seismic tomography, SIAM Journal on Scientific Computing, 37 (2015), 334-364.  doi: 10.1137/140968331. [5] N. Buong and N. Anh, An implicit iteration method for variational inequalities over the set of common fixed points for a finite family of nonexpansive mappings in Hilbert spaces, Fixed Point Theory and Applications, 1 (2011), Art. ID 276859, 10 pp. [6] J. Carey and D. Zilberman, A model of investment under uncertainty: Modern irrigation technology and emerging markets in water, American Journal of Agricultural Economics, 84 (2002), 171-183.  doi: 10.1111/1467-8276.00251. [7] S. Chang, J. Wang and X. Wang, A fitted finite volume method for real option valuation of risks in climate change, Computers and Mathematics with Applications, 70 (2015), 1198-1219.  doi: 10.1016/j.camwa.2015.07.003. [8] S. Chang and X. Wang, Modelling and computation in the valuation of carbon derivatives with stochastic convenience yields, Plos One, 10 (2015), e0125679.  doi: 10.1371/journal.pone.0125679. [9] S. Chang, X. Wang and Z. Wang, Modeling and computation of transboundary industrial pollution with emissions permits trading by stochastic differential game, PLoS ONE, 10 (2015), e0138641.  doi: 10.1371/journal.pone.0138641. [10] L. Chorn and S. Shokhor, Real options for risk management in petrolem development investments, Energy Economics, 28 (2006), 489-505. [11] B. Diomande and A. Zalinescu, Maximum principle for an optimal control problem associated to a stochastic variational inequality with delay, Electronic Journal of Probability, 20 (2014), 1-35.  doi: 10.1214/EJP.v20-2741. [12] A. Dixit and R. Pindyck, Investment under Uncertainty, Princeton University Press, Princeton, 1994. [13] R. France, Exploring the bonds and boundaries of water management in a global context, Journal of Cleaner Production, 60 (2013), 1-3.  doi: 10.1016/j.jclepro.2013.07.004. [14] W. Han and B. Reddy, On the finite element method for mixed variational inequalities arising in elastoplasticity, SIAM Journal on Numerical Analysis, 32 (1995), 1778-1807.  doi: 10.1137/0732081. [15] Y. He, Real Options in the Energy Markets, Ph. D Thesis, University of Twente, 2007. [16] C. Huang, C. Hung and S. Wang, A fitted finite volume method for the valuation of options on assets with stochastic volatilities, Computing, 77 (2006), 297-320.  doi: 10.1007/s00607-006-0164-4. [17] C. Huang and S. Wang, A power penalty approach to a nonlinear complementarity problem, Operations Research Letters, 38 (2010), 72-76.  doi: 10.1016/j.orl.2009.09.009. [18] K. Ito and K. Kunisch, Parabolic variational inequalities: The Lagrange multiplier approach, J. Math. Pures Appl., 85 (2006), 415-449.  doi: 10.1016/j.matpur.2005.08.005. [19] L. Kobari, S. Jaimungal and Y. Lawryshyn, A real options model to evaluate the effect of environmental policies on the oil sands rate of expansion, Energy Economics, 45 (2014), 155-165.  doi: 10.1016/j.eneco.2014.06.010. [20] R. Leveque, Finite Volume Methods for Hyperbolic Problems, Cambridge University Press, Cambridge, 2002.  doi: 10.1017/CBO9780511791253. [21] J. Liu, L. Mu and X. Ye, An adaptive discontinuous finite volume method for elliptic problems, Journal of Computational and Applied Mathematics, 235 (2011), 5422-5431.  doi: 10.1016/j.cam.2011.05.051. [22] A. McClintock, Investment in Irrigation Technology: Water Use Change, Public Policy and Uncertainty, Cooperative Research Centre for Irrigation Futures, Technical Report, 2014. [23] D. Pimentel, Water resources: Agriculture, the environment, and society, BioScience, 47 (1997), 97-106.  doi: 10.2307/1313020. [24] J. Reyes and K. Kunisch, A semi-smooth Newton method for regularized state-constrained optimal control of the Navier-Stokes Equations, Computing, 78 (2006), 287-309.  doi: 10.1007/s00607-006-0183-1. [25] J. Reyes and M. Hintermuller, A duality based semismooth Newton framework for solving variational inequalities of the second kind, Interfaces and Free Boundaries, 13 (2011), 437-462.  doi: 10.4171/IFB/267. [26] P. Samuelson, Proof that properly anticipated prices fluctuate randomly, The World Scientific Handbook of Futures Markets, 6 (2015), 25-38.  doi: 10.1142/9789814566926_0002. [27] S. Wang, A novel fitted finite volume method for the Black-Scholes equation governing option pricing, IMA Journal of Numerical Analysis, 24 (2004), 699-720.  doi: 10.1093/imanum/24.4.699. [28] Y. Wang, X. Chang, Z. Chen, Y. Zhong and T. Fan, Impact of subsidy policies on recycling and remanufacturing using system dynamics methodology: a case of auto parts in China, Journal of Cleaner Production, 74 (2014), 161-171.  doi: 10.1016/j.jclepro.2014.03.023. [29] T. Wang and R. Neufville, Building real options into physical systems with stochastic mixed-integer programming, In 8th Annual Real Options International Conference, (2004), 23-32. [30] G. Wang and X. Yang, The regularization method for a degenerate parabolic variational inequality arising from American option valuation, International Journal of Numerical Analysis and Modeling, 5 (2008), 222-238. [31] S. Wang and X. Yang, A power penalty method for linear complementarity problems, Operations Research Letters, 36 (2008), 211-214.  doi: 10.1016/j.orl.2007.06.006. [32] S. Wang, X. Yang and K. Teo, Power penalty method for a linear complementarity problem arising from American option valuation, Journal of Optimization Theory and Applications, 129 (2006), 227-254.  doi: 10.1007/s10957-006-9062-3. [33] S. Wang, S. Zhang and Z. Fang, A superconvergent fitted finite volume method for BlackScholes equations governing European and American option valuation, Numerical Methods for Partial Differential Equations, 31 (2015), 1190-1208.  doi: 10.1002/num.21941. [34] A. Wasylewicz, Analysis of the power penalty method for American options using viscosity solutions, Thesis, University of Oslo, 2008. [35] S. Xie, H. Xu and H. Huang, Some iterative numerical methods for a kind of system of mixed nonlinear variational inequalities, Journal of Mathematics Research, 6 (2014), 65-69.  doi: 10.5539/jmr.v6n1p65. [36] A. Zalinescu, Stochastic variational inequalities with jumps, Stochastic Processes and their Applications, 124 (2014), 785-811.  doi: 10.1016/j.spa.2013.09.005. [37] S. Zhang, X. Wang and A. Shananin, Modeling and computation of mean field equilibria in producers' game with emission permits trading, Communications in Nonlinear Science and Numerical Simulation, 37 (2016), 238-248.  doi: 10.1016/j.cnsns.2016.01.020. [38] K. Zhang, S. Wang, X. Yang and K. Teo, A power penalty approach to numerical solutions of two-asset American options, Numerical Mathematics: Theory, Method and Applications, 2 (2009), 202-233. [39] [40] [41]
European real option values for Test 1
American real option values for Test 2
The Δ and the optimal exercise boundary for Test 2
Comparative statics for the main parameters for Test 2
Real option values for Test 3
The ∆ and the optimal exercise boundary for Test 3
The effect of the suspending operation on investment boundary for Test 3
Trigger prices under different levels of subsidy in four time points
Computed errors in the $L^{\infty}$-norm at $t = 0$
 mesh $L^{\infty}$-norm ratio mesh $L^{\infty}$-norm ratio} $2^5\times2^4$ 197.4574 $2^{10}\times2^9$ 0.4418 1.9276 $2^6\times2^5$ 55.1497 3.5804 $2^{11}\times2^{10}$ 0.0453 9.7528 $2^7\times2^6$ 28.7732 1.9176 $2^{12}\times2^{11}$ 0.0124 3.6532 $2^8\times2^7$ 2.9357 9.8012 $2^{13}\times2^{12}$ 0.0054 2.2963 $2^9\times2^8$ 0.8512 3.4489
 mesh $L^{\infty}$-norm ratio mesh $L^{\infty}$-norm ratio} $2^5\times2^4$ 197.4574 $2^{10}\times2^9$ 0.4418 1.9276 $2^6\times2^5$ 55.1497 3.5804 $2^{11}\times2^{10}$ 0.0453 9.7528 $2^7\times2^6$ 28.7732 1.9176 $2^{12}\times2^{11}$ 0.0124 3.6532 $2^8\times2^7$ 2.9357 9.8012 $2^{13}\times2^{12}$ 0.0054 2.2963 $2^9\times2^8$ 0.8512 3.4489
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